Question

A ray of light from a denser medium strikes a rarer medium at angle of incidence $i$. The reflected and refracted rays make an angle of $90^{\circ }$ with each other. The angles of reflection and refraction are $r$ and $r^{\prime }$ respectively. The critical angle is

• A. ${\sin }^{-1}\left(\tan r^{\prime }\right)$
• B. ${\sin }^{-1}(\tan r)$
• C. ${\tan }^{-1}\left(\tan r^{\prime }\right)$
• D. ${\tan }^{-1}(\tan i)$

Answer 1

Answer: (B)

At the critical angle, $\sin C=\frac{1}{\mu }=\frac{1}{\sin r^{\prime }/\sin i}=\frac{\sin i}{\sin r^{\prime }}$
As is clear as shown in figure

$\angle CBD=90^{\circ }$
$\therefore 90^{\circ }-r+90^{\circ }-r^{\prime }=90^{\circ }$
or $r^{\prime }=90^{\circ }-r$
$\therefore \sin C=\frac{\sin i}{\sin \left(90^{\circ }-r\right)}=\frac{\sin i}{\cos r}$
$=\frac{\sin i}{\cos i}=\tan i(\because i=r)$
$C={\sin }^{-1}(\tan i)={\sin }^{-1}(\tan r).$